Observing Extended Sources with the \Herschel SPIRE Fourier Transform Spectrometer

The Spectral and Photometric Imaging Receiver (SPIRE) on the European Space Agency's Herschel Space Observatory utilizes a pioneering design for its imaging spectrometer in the form of a Fourier Transform Spectrometer (FTS). The standard FTS data reduction and calibration schemes are aimed at objects with either a spatial extent much larger than the beam size or a source that can be approximated as a point source within the beam. However, when sources are of intermediate spatial extent, neither of these calibrations schemes is appropriate and both the spatial response of the instrument and the source's light profile must be taken into account and the coupling between them explicitly derived. To that end, we derive the necessary corrections using an observed spectrum of a fully extended source with the beam profile and the source's light profile taken into account. We apply the derived correction to several observations of planets and compare the corrected spectra with their spectral models to study the beam coupling efficiency of the instrument in the case of partially extended sources. We find that we can apply these correction factors for sources with angular sizes up to \theta_{D} ~ 17". We demonstrate how the angular size of an extended source can be estimated using the difference between the sub-spectra observed at the overlap bandwidth of the two frequency channels in the spectrometer, at 959<\nu<989 GHz. Using this technique on an observation of Saturn, we estimate a size of 17.2", which is 3% larger than its true size on the day of observation. Finally, we show the results of the correction applied on observations of a nearby galaxy, M82, and the compact core of a Galactic molecular cloud, Sgr B2.